R.SINDHI MADHURI

A.ANURAG REDDY

G.USHASWI

ROHIT UPADHYAY

Face recognition using PCA

• IDEA

• OPERATIONS

• MERITS

• DEMERITS

• APPLICATIONS

CONTENTS

PCA

Eigenfaces: the idea

Eigenvectors and Eigenvalues

Learning Eigenfaces from training sets of faces

Co-variance

Recognition and reconstruction

IDEA

PCA means Principle Component Analysis.

PCA was invented in 1901 by Karl Pearson

PCA involves the calculation of the eigenvalue decomposition of a data covariance matrix or singular value decomposition of a data matrix, usually after mean centering the data for each attribute.

PCA

Three basic steps involved in PCA are:Identification{by eigen faces}Recognition{matching eigen faces}Categorization{by grouping}

Algorithm

In Digital Image Processing, we convert 2-D images into matrix form for clear analysis.Every matrix can be represented with the help of its eigen vectors.An eigenvector is a vector that obeys the following rule:

Where A is a matrix , is a scalar (called the eigenvalue)

e.g. one eigenvector of is since

so for this eigenvector of this matrix the eigenvalue is 4

v vA

2 3

2 1

A3

2

2 3 3 12 34

2 1 2 8 2

EIGEN VECTORS

EIGEN FACESThink of a face as being a weighted combination of some “component” or “basis” faces

These basis faces are called eigen faces.

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2

1

2

N

a

a

a

2

1

2

N

b

b

b

2

1

2

N

c

c

c

2

1

2

N

d

d

d

2

1

2

N

e

e

e

2

1

2

N

f

f

f

Eigenfaces: representing faces

We compute the average face

2 2 2

1 1 1

2 2 21, 8

N N N

a b h

a b hm where M

M

a b h

Then subtract it from the training faces

2 2 2 2 2 2 2 2

2 2

1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2

1 1 1 1

2 2

, , , ,

,

m m m m

N N N N N N N N

m m

N N

a m b m c m d m

a m b m c m d ma b c d

a m b m c m d m

e m f m

e m fe f

e m

2 2 2 2 2 2

1 1 1 1

2 2 2 2 2 2, ,m m

N N N N N N

g m h m

m g m h mg h

f m g m h m

Now we build the matrix which is N2 by M

The covariance matrix which is N2 by N2

m m m m m m m mA a b c d e f g h

Cov AA

The covariance matrix has eigenvectors covariance matrix eigenvectors

eigenvalues

Eigenvectors with larger eigenvectors correspond to

directions in which the data varies more

Finding the eigenvectors and eigenvalues of the

covariance matrix for a set of data is termed principle components analysis

.617 .615

.615 .717C

1

.735

.678

2

.678

.735

1 0.049 2 1.284

1 1 2 21

1 2

( )( )cov( , )

1

ni i

i

x x x xx x

n

The covariance of two variables is:

RecognitionA face image can be projected into this face space by

pk = UT(xk – m) where k=1,…,m

To recognize a face

2

1

2

N

r

r

r

2 2

1 1

2 2

m

N N

r m

r mr

r m

Subtract the average face from it

Compute its projection onto the face space U

mU r

Compute the distance in the face space between the face and all known faces

22 1..i i for i M

Compute the threshold 1max , 1..

2 i j for i j M

Distinguish between• If then it’s not a face; the

distance between the face and its reconstruction is larger than threshold

• If then it’s a new face

• If then it’s a known face because the distance in the face space between the face and all known faces is larger than threshold

min iand , ( 1.. )iand i M

RECONSTRUCTIONImage is reconstructed in the 3rd case, if , ( 1.. )iand i M

Using the MATLAB code, original image and reconstructed image are shown.

Ex:

MERITS

Relatively simpleFastRobustExpression

- Change in feature location and shape.

DEMERITSVariations in lighting conditions

Different lighting conditions for enrolment and query. Bright light causing image saturation.

APPLICATIONS:Various potential applications, such as

• Person identification. • Human-computer

interaction.• Security systems.

Thank You